is implied for any \(z_{mn}=y_{mn}exp(i\theta _{mn})\)  with
\(y_{mn}\ge 0\)  and \(\sum _{mn}y_{mn}^2=1\) , where
\(A_{mn}=M^{\frac{1}{2}}\Psi ^T S_{mn}\Psi M^{\frac{1}{2}}\) . Since
\(\sum _{mn}z_{mn}A^{mn}\)  is a symmetric matrix, we can always find a
unitary matrix \(U\)  such that
\(\sum _i|U^T(\sum _{mn}z_{mn}A_{mn})U|_{ii}=\Vert \sum _{mn}z_{mn}A^{mn}\Vert \) 
as shown in Ref. [1]}, where \(\Vert \cdot \Vert \)  stands for the trace
norm defined by \(\Vert G\Vert =\textrm {Tr}(GG^\dag )^{1/2}\) . For an arbitrary
unitary matrix \(V\) , we have
\(&&\sum _i|V^T(\sum _{mn}z_{mn}A_{mn})V|_{ii}\\&\!\!=\!\!&\sum _i|V^T(U^{-1})^TU^T(\sum _{mn}z_{mn}A_{mn})UU^{-1}V|_{ii}\\&\!\!=\!\!&\sum _i|V^T(U^{-1})^TDiag(\lambda _1,\lambda _2\cdots )U^{-1}V|_{ii}\\&\!\!\le \!\!&\sum _{ij}|(U^{-1}V)_{ij}|^2\lambda _i\\&\!\!=\!\!&\sum _i\lambda _i,\) 
